Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{k^2 - 4}{k + 2}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = k$ $ b = \sqrt{4} = 2$ So we can rewrite the expression as: $a = \dfrac{({k} + {2})({k} {-2})} {k + 2} $ We can divide the numerator and denominator by $(k + 2)$ on condition that $k \neq -2$ Therefore $a = k - 2; k \neq -2$